Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in $KK$-theory, we endow the $K$-groups of the arithmetic manifold $X / \Gamma$, of the reduced group $C^*$-algebra $C^*_r(\Gamma)$ and of the boundary crossed product algebra $C(\partial X) \rtimes\Gamma$ with Hecke operators. In the case when $\Gamma$ is a group of real hyperbolic isometries, the $K$-theory and $K$-homology groups of these $C^{*}$-algebras are related by a Gysin six-term exact sequence and we prove that this Gysin sequence is Hecke equivariant. Finally, when $\Gamma$ is a Bianchi group, we assign explicit unbounded Fredholm modules (i.e. spectral triples) to (co)homology classes, inducing Hecke-equivariant isomorphisms between the integral cohomology of $\Gamma$ and each of these $K$-groups. Our methods apply to case $\Gamma \subset \mathbf {PSL}(\mathbf Z)$ as well.

In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of $\mathbf H/\Gamma$ to a spectral triple on the purely infinite geodesic boundary crossed product algebra $C(\partial \mathbf H) \rtimes\Gamma$.

中文翻译：

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Hecke算子在$ KK $理论和Bianchi组的$ K $同源性中

令$ \ Gamma $是作用于其关联的全局对称空间$ X $的无扭转算术组。假定$ X $是非紧凑类型，并且让$ \ Gamma $作用于测地边界$ \ $ X $的局部X $。通过$ KK $-理论的一般构造，我们赋予算术流形$ X / \ Gamma $的$ K $-组，化简组$ C ^ * $-代数$ C ^ * _ r（\ Gamma）$的$ K $-组。和带有Hecke算子的边界乘积代数$ C（\ partial X）\ rtimes \ Gamma $。在$ \ Gamma $是一组实双曲等式的情况下，这些$ C ^ {*} $代数的$ K $理论和$ K $同源性组由Gysin六项精确序列关联并且证明了该Gysin序列是Hecke等变的。最后，当$ \ Gamma $是Bianchi组时，我们将显式无界Fredholm模块（即光谱三元组）分配给（共）同构类，诱导$ \ Gamma $的积分同调和每个$ K $-组之间的Hecke等价同构。我们的方法也适用于大小写$ \ Gamma \ subset \ mathbf {PSL}（\ mathbf Z）$。

特别是，我们使用无界的Kasparov乘积来推动Dirac算子在$ \ mathbf H / \ Gamma $的Borel-Serre压缩中的嵌入表面达到纯无限地线边界交叉乘积代数$ C（\ partial \ mathbf H）\ rtimes \ Gamma $。